Exponential Functions in Cartesian Differential Categories

Abstract

In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $$e^x$$ e x from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of $$e^0 = 1$$ e 0 = 1 , $$e^{x+y} = e^x e^y$$ e x + y = e x e y , and $$\frac{\partial e^x}{\partial x} = e^x$$ ∂ e x ∂ x = e x all hold. Every differential exponential map induces a commutative rig, which we call a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. In particular, differential exponential maps can be defined without the need of limits, converging power series, or unique solutions of certain differential equations—which most Cartesian differential categories do not necessarily have. That said, we do explain how every differential exponential map does provide solutions to certain differential equations, and conversely how in the presence of unique solutions, one can derivative a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, the split complex exponential function, and the dual numbers exponential function. As another source of interesting examples, we also study differential exponential maps in the coKleisli category of a differential category.

Cartesian Differential Comonads and New Models of Cartesian Differential Categories

“…The differential combinator satisfies seven axioms, known as [CD.1] to [CD.7], which formalize the basic identities of the (total) derivative from multi-variable differential calculus such as the chain rule, linearity in vector argument, symmetry of partial derivatives, etc. Cartesian differential categories have been able to formalize various concepts of differential calculus such as solving differential equations [17], de Rham cohomology [25], exponential functions [39], tangent bundles [15,42], and integration [19]. Cartesian differential categories have also found applications throughout computer science, specifically for the foundations of differentiable programming languages [1,23], as well as for machine learning and automatic differentiation [24,47].…”

Combining fixpoint and differentiation theory

“…In this background section, we review Cartesian differential categories. For a more in-depth introduction, we refer the reader to [8,14,32,46,48].…”

Taylor Expansion as a Monad in Models of DiLL

“…As mentioned above, we will explain below why codigging should be interpreted as a generalized expentional function. To help justify this claim, let us first quickly review the generalization of the exponential function e x in context of differential storage categories, called a !-differential exponential map, which was introduced by Lemay in [12]. Classically, e x admits numerous equivalent characterization either as the inverse of the natural logarithm function, or as a limit or converging power series, or even as the unique solution to a differential equation.…”

“…In particular, we have exp(exp(ϕ)) = p A (exp(δ ϕ )). Lastly, equation (12) states the interaction between p and d, which we call the "cochain rule", since the compatibility between d and p is the chain rule. On the one hand,…”

“…To justify this last claim, we must consider what e x in the context of DiLL would even be. The answer to this question comes from an independent categorical exploration by Lemay [12]. We will explain how the axioms of codigging precisely state that p fits in this categorical axiomatization of exponential map.…”